exponential map

Let X and Y be n×n complex matrices and let a and b be arbitrary complex numbers. We denote the n×n identity matrix by I and the zero matrix by 0. The matrix exponential satisfies the following properties: e0 = I. eaXebX = e(a + b)X. e< ...

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Matrix exponential, Matrix exponential - Properties, Matrix exponential - Linear differential equations, Matrix exponential - The exponential of sums, Matrix exponential - The exponential map, Matrix exponential - Computing the matrix exponential, Matrix exponential - Diagonalizable case, Matrix exponential - Nilpotent case, Matrix exponential - General case, Matrix exponential - Calculations, Matrix exponential - Applications, Matrix exponential - Linear differential equations

Read more here: » Matrix exponential: Encyclopedia II - Matrix exponential - Properties

On a (pseudo-)Riemannian manifold M a geodesic is defined as a smooth curve γ(t) that parallel transports its own tangent vector. That is, . where ∇ stands for the Levi-Civita connection on M. In the case of a Riemannian manifold, the geodesics that one obtains this way are identical to geodesics for the induced metric space. In terms of local coordinates on M the geodesic equation can be written (using the summation convention): where xa(t) are the coordinates of t ...

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Geodesic, Geodesic - Introduction, Geodesic - Examples, Geodesic - Metric geometry, Geodesic - pseudo-Riemannian geometry, Geodesic - Existence and uniqueness, Geodesic - Geodesic flow, Geodesic - Geodesic spray, Geodesic - Variations and generalizations

Read more here: » Geodesic: Encyclopedia II - Geodesic - pseudo-Riemannian geometry

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. Vector fields are often used in physics to model for example the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point. In the rigorous mathematical treatment, (tangent) vector fields are defined on manifolds as sections of the manifold's tangent bundle. V ...

Including:

• Vector field - Definition
• Vector field - Notes
• Vector field - Examples
• Vector field - Gradient field
• Vector field - Central field
• Vector field - Curve integral
• Vector field - Flow curves
• Vector field - Difference between scalar and vector field
• Vector field - Example 3

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The shortest path between two points in a curved space can be found by writing the equation for the length of a curve, and then minimizing this length using standard techniques of calculus and differential equations. Equivalently, a different quantity may be defined, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic. Intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its length, and in so doing will minimize its energy; the r ...

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Geodesic, Geodesic - Introduction, Geodesic - Examples, Geodesic - Metric geometry, Geodesic - pseudo-Riemannian geometry, Geodesic - Existence and uniqueness, Geodesic - Geodesic flow, Geodesic - Geodesic spray, Geodesic - Variations and generalizations

Read more here: » Geodesic: Encyclopedia II - Geodesic - Introduction

Matrix exponential - Linear differential equations. The matrix exponential has applications to systems of linear differential equations. Recall that a differential equation of the form y′ = Cy has solution eCx. If we consider the vector we can express a system of coupled linear differential equations as If we make an ansatz and use an integrating factor of e−Ax and multiply t ...

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Matrix exponential, Matrix exponential - Properties, Matrix exponential - Linear differential equations, Matrix exponential - The exponential of sums, Matrix exponential - The exponential map, Matrix exponential - Computing the matrix exponential, Matrix exponential - Diagonalizable case, Matrix exponential - Nilpotent case, Matrix exponential - General case, Matrix exponential - Calculations, Matrix exponential - Applications, Matrix exponential - Linear differential equations

Read more here: » Matrix exponential: Encyclopedia II - Matrix exponential - Applications

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. 3-sphere - Explanation. An ordinary sphere, or 2-sphere, consists of all points equidistant from a single point in ordinary 3-dimensional Euclidean space, R3. A 3-sphere consists of all points equidistant from a single point in R4. Whereas a 2-sphere is a smooth 2-dimensional surface, a 3-sphere is an object with three dimensions, also known as 3-manifold. In an entirely analogous manner one c ...

Including:

• 3-sphere - Explanation
• 3-sphere - Definition
• 3-sphere - Elementary properties
• 3-sphere - Topological construction
• 3-sphere - Unslicing
• 3-sphere - Unpuncturing
• 3-sphere - Topological properties
• 3-sphere - Coordinate systems on the 3-sphere
• 3-sphere - Hyperspherical coordinates
• 3-sphere - Hopf coordinates
• 3-sphere - Stereographic coordinates
• 3-sphere - Group structure
• 3-sphere - Tangents
• 3-sphere - In literature
• 3-sphere - Related topics
• 3-sphere - External link

Read more here: » 3-sphere: Encyclopedia - 3-sphere

Consider the matrix which has Jordan form and transition matrix Now, and So, The exponential calculation for a 1×1 matrix is clearly trivial, with eJ1(4)=e4 so, Clearly, to calculate the Jordan form and to evaluate the exponential this way is very tedious. Often, it will often suffice to calculate the action of the exponential matrix upon some vector in applications, and ...

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Matrix exponential, Matrix exponential - Properties, Matrix exponential - Linear differential equations, Matrix exponential - The exponential of sums, Matrix exponential - The exponential map, Matrix exponential - Computing the matrix exponential, Matrix exponential - Diagonalizable case, Matrix exponential - Nilpotent case, Matrix exponential - General case, Matrix exponential - Calculations, Matrix exponential - Applications, Matrix exponential - Linear differential equations

Read more here: » Matrix exponential: Encyclopedia II - Matrix exponential - Calculations

In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve γ: I → M from the unit interval I to the metric space M is a geodesic if there is a constant v ≥ 0 such that for any t ∈ I there is a neighborhood J of t in I such that for any t1, t2See also:

Geodesic, Geodesic - Introduction, Geodesic - Examples, Geodesic - Metric geometry, Geodesic - pseudo-Riemannian geometry, Geodesic - Existence and uniqueness, Geodesic - Geodesic flow, Geodesic - Geodesic spray, Geodesic - Variations and generalizations

Read more here: » Geodesic: Encyclopedia II - Geodesic - Metric geometry

Matrix exponential - Diagonalizable case. If a matrix is diagonal: then its exponential can be obtained by just exponentiating every entry on the main diagonal: This also allows one to exponentiate diagonalizable matrices. If A = UDU−1 and D is diagonal, then eA = UeDU−1See also:

Matrix exponential, Matrix exponential - Properties, Matrix exponential - Linear differential equations, Matrix exponential - The exponential of sums, Matrix exponential - The exponential map, Matrix exponential - Computing the matrix exponential, Matrix exponential - Diagonalizable case, Matrix exponential - Nilpotent case, Matrix exponential - General case, Matrix exponential - Calculations, Matrix exponential - Applications, Matrix exponential - Linear differential equations

Read more here: » Matrix exponential: Encyclopedia II - Matrix exponential - Computing the matrix exponential

We define the kernel of h to be ker(h) = { u in G : h(u) = eH } and the image of h to be im(h) = { h(u) : u in G }. The kernel is a normal subgroup of G (in fact, h(g-1 u g) = h(g)-1 h(u) h(g) = h(g)-1 eH h(g) = h(g ...

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Group homomorphism, Group homomorphism - Image and kernel, Group homomorphism - Examples, Group homomorphism - The category of groups, Group homomorphism - Isomorphisms endomorphisms and automorphisms, Group homomorphism - Homomorphisms of abelian groups

Read more here: » Group homomorphism: Encyclopedia II - Group homomorphism - Image and kernel

Vector field - Gradient field. Vector fields can be constructed out of scalar fields using the vector operator gradient which gives rise to the following definition. A vector field V over S is called a gradient field or a conservative field if there exists a real valued function f on X(a scalar field) such that The path integral along any closed curve γ (γ(0) = γ(1)) in a gradient field is zero. Vector field - C ...

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Vector field, Vector field - Definition, Vector field - Notes, Vector field - Examples, Vector field - Gradient field, Vector field - Central field, Vector field - Curve integral, Vector field - Flow curves, Vector field - Difference between scalar and vector field, Vector field - Example 3

Read more here: » Vector field: Encyclopedia II - Vector field - Examples

Sums and scalar products of skew-symmetric matrices are again skew-symmetric. Hence, the skew-symmetric matrices form a vector space. If matrices A and B are both skew-symmetric, then their product AB is a symmetric matrix. On the other hand, the triple product BTAB is skew-symmetric. The "skew-symmetric component" of a matrix A is the matrix B = (A − AT)/2; the "symmetric component" of A is C = (A + AT)/2; the matrix A is the ...

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Skew-symmetric matrix, Skew-symmetric matrix - Properties, Skew-symmetric matrix - The determinant of a skew-symmetric matrix, Skew-symmetric matrix - Spectral theory, Skew-symmetric matrix - Alternating forms, Skew-symmetric matrix - Infinitesimal rotations

Read more here: » Skew-symmetric matrix: Encyclopedia II - Skew-symmetric matrix - Properties

Using the natural logarithm, one can define more general exponential functions. The function defined for all a > 0, and all real numbers x, is called the exponential function with base a. Note that the equation above holds for a = e, since Exponential functions "translate between addition and multiplication" as is expressed in the following exponential laws: ...

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Exponential function, Exponential function - Properties, Exponential function - Derivatives and differential equations, Exponential function - Formal definition, Exponential function - Numerical value, Exponential function - On the complex plane, Exponential function - Matrices and Banach algebras, Exponential function - On Lie algebras, Exponential function - Double exponential function

Read more here: » Exponential function: Encyclopedia II - Exponential function - Properties

Curvature of Riemannian manifolds - The curvature tensor. The curvature of Riemannian manifold can be described in various ways; the most standard one is the curvature tensor, given in terms of a Levi-Civita connection (or covariant differentiation) and Lie bracket [ * , * ] by the following formula: Here R(u,v) is a linear transformation of the tangent space of the manifold; it is linear in each argument. ...

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Curvature of Riemannian manifolds, Curvature of Riemannian manifolds - Ways to express the curvature of a Riemannian manifold, Curvature of Riemannian manifolds - The curvature tensor, Curvature of Riemannian manifolds - Sectional curvature, Curvature of Riemannian manifolds - Curvature form, Curvature of Riemannian manifolds - The curvature operator, Curvature of Riemannian manifolds - Further curvature tensors, Curvature of Riemannian manifolds - Scalar curvature, Curvature of Riemannian manifolds - Ricci curvature, Curvature of Riemannian manifolds - Weyl curvature tensor, Curvature of Riemannian manifolds - Calculation of curvature

Read more here: » Curvature of Riemannian manifolds: Encyclopedia II - Curvature of Riemannian manifolds - Ways to express the curvature of a Riemannian manifold

There are more general Heisenberg groups Hn. We begin by discussing the Real Heisenberg group of dimension 2n+1, for any integer n ≥ 1. As a group of matrices, Hn (or Hn(R) to indicate this is the Heisenberg group over the ring R) is defined as the group of square matrices of size n+2 with entries in R: where a is a row vector of length n, b is a column vector of length n and See also:

Heisenberg group, Heisenberg group - Examples, Heisenberg group - General Heisenberg group, Heisenberg group - The connection with the Weyl algebra, Heisenberg group - Weyl's view of quantum mechanics, Heisenberg group - As a sub-Riemannian manifold

Read more here: » Heisenberg group: Encyclopedia II - Heisenberg group - General Heisenberg group

Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups. Given a Lie group, a Lie algebra can be associated to it either by endowing the tangent space to the identity with the differential of the adjoint map, or by considering the left-invariant vector fields as mentioned in the examples. This association is functorial, meaning that homomorphisms of Lie groups lift to homomorphisms of Lie algebras, and various properties are satisfied by this lifting: it commutes with composition, it maps subgroups, kernels, quotients and cokernels of Lie groups to subalgebras, kernels, ...

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Lie algebra, Lie algebra - Definition, Lie algebra - Examples, Lie algebra - Homomorphisms subalgebras and ideals, Lie algebra - Relation to Lie groups, Lie algebra - Classification of Lie algebras, Lie algebra - Category theoretic definition

Read more here: » Lie algebra: Encyclopedia II - Lie algebra - Relation to Lie groups

Two convenient constructions for the topologist are the reverse of "slicing in half" and "puncturing". 3-sphere - Unslicing. A 3-sphere can be constructed topologically by "gluing" together the boundaries of a pair of 3-balls. The boundary of a 3-ball is a 2-sphere, and these two 2-spheres are to be identified. That is, imagine a pair of 3-balls of the same size, then superpose them so that their 2-spherical boundaries match, and let matching pairs of points on the pair ...

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3-sphere, 3-sphere - Explanation, 3-sphere - Definition, 3-sphere - Elementary properties, 3-sphere - Topological construction, 3-sphere - Unslicing, 3-sphere - Unpuncturing, 3-sphere - Topological properties, 3-sphere - Coordinate systems on the 3-sphere, 3-sphere - Hyperspherical coordinates, 3-sphere - Hopf coordinates, 3-sphere - Stereographic coordinates, 3-sphere - Group structure, 3-sphere - Tangents, 3-sphere - In literature, 3-sphere - Related topics, 3-sphere - External link

Read more here: » 3-sphere: Encyclopedia II - 3-sphere - Topological construction

The skew-symmetric n×n matrices form a vector space of dimension n(n − 1)/2. This is the tangent space to the orthogonal group O(n) at the identity matrix. In a sense, then, skew-symmetric matrices can be thought of as infinitesimal rotations. Another way of saying this is that the space of skew-symmetric matrices forms the Lie algebra o(n) of the Lie group O(n). The Lie bracket on this space is given by the commutator: [ASee also:

Skew-symmetric matrix, Skew-symmetric matrix - Properties, Skew-symmetric matrix - The determinant of a skew-symmetric matrix, Skew-symmetric matrix - Spectral theory, Skew-symmetric matrix - Alternating forms, Skew-symmetric matrix - Infinitesimal rotations

Read more here: » Skew-symmetric matrix: Encyclopedia II - Skew-symmetric matrix - Infinitesimal rotations

Real and complex Lie algebras can be classified to some extent, and this classification is an important step toward the classification of Lie groups. Every finite-dimensional real or complex Lie algebra arises as the Lie algebra of unique real or complex simply connected Lie group (Ado's theorem), but there may be more than one group, even more than one connected group, giving rise to the same algebra. For instance, the groups SO(3) (3×3 orthogonal matrices of determinant 1) and SU(2) (2×2 unitary matrices of determinant 1) both give rise to the sam ...

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Lie algebra, Lie algebra - Definition, Lie algebra - Examples, Lie algebra - Homomorphisms subalgebras and ideals, Lie algebra - Relation to Lie groups, Lie algebra - Classification of Lie algebras, Lie algebra - Category theoretic definition

Read more here: » Lie algebra: Encyclopedia II - Lie algebra - Classification of Lie algebras

Using the language of category theory, a Lie algebra can be defined as an object A in the category of vector spaces together with a morphism such that where and σ is the cyclic permutation braiding . In diagrammatic form: ...

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Lie algebra, Lie algebra - Definition, Lie algebra - Examples, Lie algebra - Homomorphisms subalgebras and ideals, Lie algebra - Relation to Lie groups, Lie algebra - Classification of Lie algebras, Lie algebra - Category theoretic definition

Read more here: » Lie algebra: Encyclopedia II - Lie algebra - Category theoretic definition